Is there such a thing as a nonparametric regression?
The use of a ranked response in regressions has been proposed by various authors for situations where distributional assumptions such as normality of residuals, homogeneity of variance and linearity of predictor to response do not hold.
Field, Miles and Field (2012) illustrate a robust regression procedure involving bootstrapping which can be used in an early version of R (but please note does not work in at least some later versions).
References
Conover WJ, Iman RL (1981) Rank transformations as a bridge between parametric and nonparametric statistics. The American Statistician 35(3) 124-129.
Field A, Miles J and Field Z (2012) Discovering Statistics Using R. Sage:London.
Huitema BE (1980) The ANCOVA and alternatives (1980) Chapter 12.
A technique called RGLM which minimizes the ranks of the residuals in the regression is available in MINITAB using the RREGR procedure and mentioned in the reference below which compares it to a regression using a ranked response. They recommend RGLM be used for situations involving more than one predictor variable.
McKean JW and Vidmar TJ (1994) A comparison of two rank-based methods for the analysis of linear models The American Statistician 48(3) 220-229.
Quade D (1967) Rank analysis of covariance Journal of the American Statistical Association 62 1187-1200.
Shirley EAC (1979) A distribution-free method for analysis of covariance based on ranked data. Applied Statistics 30(2) 158-162.